Transformation Groups in Differential Geometry

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Springer Science & Business Media, 1995/02/15 - 182 ページ
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Given a mathematical structure, one of the basic associated mathematical objects is its automorphism group. The object of this book is to give a biased account of automorphism groups of differential geometric struc tures. All geometric structures are not created equal; some are creations of ~ods while others are products of lesser human minds. Amongst the former, Riemannian and complex structures stand out for their beauty and wealth. A major portion of this book is therefore devoted to these two structures. Chapter I describes a general theory of automorphisms of geometric structures with emphasis on the question of when the automorphism group can be given a Lie group structure. Basic theorems in this regard are presented in §§ 3, 4 and 5. The concept of G-structure or that of pseudo-group structure enables us to treat most of the interesting geo metric structures in a unified manner. In § 8, we sketch the relationship between the two concepts. Chapter I is so arranged that the reader who is primarily interested in Riemannian, complex, conformal and projective structures can skip §§ 5, 6, 7 and 8. This chapter is partly based on lec tures I gave in Tokyo and Berkeley in 1965.
 

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目次

I Automorphisms of GStructures
1
2 Examples of GStructures
5
3 Two Theorems on Differentiable Transformation Groups
13
4 Automorphisms of Compact Elliptic Structures
16
5 Prolongations of GStructures
19
6 Volume Elements and Symplectic Structures
23
2 Infinitesimal Isometries and Infinitesimal Affine Transformations
42
3 Riemannian Manifolds with Large Group of Isometries
46
9 Projectively Induced Holomorphic Transformations
106
10 Zeros of Infinitesimal Isometries
112
11 Zeros of Holomorphic Vector Fields
115
12 Holomorphic Vector Fields and Characteristic Numbers
119
IV Affine Conformal and Projective Transformations
122
2 Affine Transformations of Riemannian Manifolds
125
3 Cartan Connections
127
4 Projective and Conformal Connections
131

4 Riemannian Manifolds with Little Isometries
55
5 Fixed Points of Isometries
59
6 Infinitesimal Isometries and Characteristic Numbers
67
III Automorphisms of Complex Manifolds
77
2 Compact Complex Manifolds with Finite Automorphism Groups
82
3 Holomorphic Vector Fields and Holomorphic 1Forms
90
4 Holomorphic Vector Fields on Kiihler Manifolds
92
5 Compact EinstetnKa hler Manifolds
95
6 Compact Kahler Manifolds with Constant Scalar Curvature
97
7 Conformal Changes of the Laplacian
100
8 Compact Kahler Manifolds with Nonpositive First Chern Class
103
5 Frames of Second Order
139
6 Projective and Conformal Structures
141
7 Projective and Conformal Equivalences
145
Appendices
150
2 Some Integral Formulas
154
3 Laplacians in Local Coordinates
157
4 A Remark on d dCohomology
159
Bibliography
160
Index
181
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著者について (1995)

Biography of Shoshichi Kobayashi

Shoshichi Kobayashi was born January 4, 1932 in Kofu, Japan. After obtaining his mathematics degree from the University of Tokyo and his Ph.D. from the University of Washington, Seattle, he held positions at the Institute for Advanced Study, Princeton, at MIT and at the University of British Columbia between 1956 and 1962, and then moved to the University of California, Berkeley, where he is now Professor in the Graduate School.

Kobayashi's research spans the areas of differential geometry of real and complex variables, and his numerous resulting publications include several book: Foundations of Differential Geometry with N. Nomizu, Hyperbolic Complex Manifolds and Holomorphic mappings and Differential Geometry of Complex Vector Bundles.

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